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Place a rack in the top third of the oven and preheat the oven to broil.

Trim off about an inch from the bottom of the asparagus and peel about halfway up. Toss the asparagus with the olive oil on a baking sheet. Season with salt and pepper and broil until slightly blistered and tender, about 7 minutes. Remove from the oven and set aside.

Combine the mayonnaise, Parmesan, lemon juice and 1/2 teaspoon pepper and set aside.

Fill a straight-sided skillet with about 2 inches of water and bring it to a bare simmer over medium heat. Add the vinegar. Crack an egg into a cup and gently slip it into the water. Repeat with the remaining eggs and cook until the whites are set but the yolks are still runny, 3 to 4 minutes. Using a slotted spoon, gently scoop the eggs from the water and blot dry with a paper towel.

To serve, divide the asparagus among 4 plates, top each with a poached egg, drizzle with a quarter of the sauce and garnish with a sprig of dill.